Capacitor

Linear or nonlinear capacitor with optional tolerance, operational limits and fault behavior

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Libraries:
Simscape / Electrical / Passive

Description

The Capacitor block lets you model linear, nonlinear (table-based), and frequency-dependent capacitors, including polar capacitors. Optionally, you can also model the following effects:

• Tolerances

• Operating Limits

• Faults

You can turn these modeling options on and off independently of each other. When all the additional options are turned off, the component behavior is identical to the Simscape™ Foundation library Capacitor block.

In its simplest form, the Capacitor block models a linear capacitor, described with the following equation:

$$I=C\frac{dV}{dt}$$

where:

• I is the current.

• C is the capacitance.

• V is the voltage.

• t is the time.

To model a nonlinear or polar capacitor, set the Capacitance model parameter to Lookup table and provide a lookup table of capacitance-voltage values:

• For polar capacitors, where this lookup table is asymmetric with respect to the applied terminal voltage, set the Symmetric C-V table parameter to No - use C-V data as-is.

• For other types of nonlinear capacitor, ensure symmetry of the capacitance with regards to the applied terminal voltage by setting the Symmetric C-V table parameter to Yes - use voltage magnitude when computing C.

To model a frequency-dependent capacitor with resistive and dielectric losses, set the Capacitance model parameter to Dielectric relaxation (Debye). The Debye relaxation model considers a collection of noninteracting dipoles in the frequency domain. The result is in terms of a complex permittivity. The real (𝜖′) and imaginary (𝜖′′) parts of the complex permittivity are given by the equations:

$$\epsilon \text{'}={\epsilon }_{\infty }+\frac{{\epsilon }_s-{\epsilon }_{\infty }}{1+{\omega }^2{\tau }^2}$$

$$\epsilon \text{'}\text{'}=\frac{({\epsilon }_s-{\epsilon }_{\infty })\omega \tau }{1+{\omega }^2{\tau }^2},$$

where 𝜔 is the radial frequency, 𝜖_∞ is the real permittivity at very high frequency, 𝜖_𝑠 is the real permittivity at low frequency and 𝜏 is the relaxation time constant.

In the time-domain, the characteristic equation for a capacitor in the Debye model is

$$\tau i=-q+C_s(\alpha \tau \dot{\nu }+\nu ),$$

where 𝐶_𝑠 is the low-frequency capacitance, 𝛼 = 𝜖_∞/𝜖_𝑠, 𝑞 is the charge, 𝑖 is the current and 𝑣 is the voltage across the capacitor.

Most datasheets do not provide explicit values for the complex permittivity nor for the relaxation time; however, often the dissipation factor is provided at two frequencies. The parameters 𝛼 and 𝜏 may be derived from these values as described by the equations:

$$\tau =\frac{D{F}_{1}D{F}_2({\omega }_2^2-{\omega }_1^2)+\sqrt{D{F}_1^{2}D{F}_2^2{({\omega }_2^2-{\omega }_1^2)}^2+4{\omega }_1{\omega }_2(D{F}_2{\omega }_2-D{F}_1{\omega }_1)(D{F}_1{\omega }_2-D{F}_2{\omega }_1)}}{2{\omega }_1{\omega }_2(D{F}_2{\omega }_2-D{F}_1{\omega }_1)}$$

$$\alpha =\frac{{\omega }_1\tau -D{F}_1}{{\omega }_1\tau (1+D{F}_1{\omega }_1\tau )},$$

where 𝜔₁ and 𝜔₂ are two different frequencies, and DF₁ and DF₂ are the dissipation factors calculated at said frequencies, respectively.

The argument of the square root must be positive for a valid parameterization in the Debye model.

Tolerances

You can apply tolerances to the nominal value you provide for the Capacitance parameter. Datasheets typically provide a tolerance percentage for a given capacitor type. The table shows how the block applies tolerances and calculates capacitance based on the selected Tolerance application option.

Option	Capacitance Value
None — use nominal value	C
Random tolerance	Uniform distribution: C · (1 – tol + 2· tol· rand) Gaussian distribution: C · (1 + tol · randn / nSigma)
Apply maximum tolerance value	C · (1 + tol )
Apply minimum tolerance value	C · (1 – tol )

In the table,

• C is the Capacitance parameter value, nominal capacitance.

• tol is the fractional tolerance, Capacitance tolerance (%) /100.

• nSigma is the value you provide for the Number of standard deviations for quoted tolerance parameter.

• rand and randn are standard MATLAB^® functions for generating uniform and normal distribution random numbers.

Note

If you choose the Random tolerance option and you are in "Fast Restart" mode, the random tolerance value is updated on every simulation if at least one between the fractional tolerance, tol, or the Number of standard deviations for quoted tolerance, nSigma, is set to Run-time and is defined with a variable (even if you do not modify that variable).

Operating Limits

You can specify operating limits in terms of maximum working voltage and the maximum (instantaneous) power dissipation in the series resistance and in the parallel conductance of the capacitor.

For polar capacitors, you can define the working voltage range in such a way that the block provides a warning, or an error, if the polarity of the applied voltage becomes incorrect.

When an operating limit is exceeded, the block can either generate a warning or stop the simulation with an error. For more information, see Operating Limits.

Model Thermal Effects

Since R2024a

You can expose the thermal port to observe the power loss and the change in temperature of the capacitor and to set the thermal mass. To expose the thermal port, in the Thermal section, select the Model thermal port parameter.

Exposing the thermal port also enables a parameter, Temperature, in the Operating Limits section, that defines the maximum valid temperature for the block operation.

Use the Variables tab to set the initial temperature target.

Faults

To model a fault in the Capacitor block, in the Faults section, click the Add fault hyperlink next to the fault that you want to model. In the Add Fault window, specify the fault properties. For more information about fault modeling, see Fault Behavior Modeling and Fault Triggering.

Instantaneous changes in capacitor parameters are unphysical. Therefore, when the Capacitor block enters the faulted state, the capacitance, resistance, and conductance transition to their faulted values over a period of time, according to the following formula:

CurrentValue = FaultedValue – (FaultedValue – UnfaultedValue) · sech(∆t / τ)

where:

• ∆t is the time since the onset of the fault condition.

• τ is the user-defined time constant associated with the fault transition.

The block can trigger the start of fault transition:

• At a specific time.

• When terminal voltage is outside the permissible voltage range for longer than a specific time interval.

If you want to trigger a fault at a specific time, in the Fault Inspector window, set Trigger type to Timed. If you want to determine whether a system fails and, if so, when it fails, in the Fault Inspector window, set Trigger type to Behavioral.

Variables

Use the Variables section of the block interface to set the priority and initial target values for the block variables prior to simulation. For more information, see Set Priority and Initial Target for Block Variables.

The Capacitor voltage variable lets you specify a high-priority target for the initial capacitor voltage at the start of simulation.

The Temperature variable lets you specify a high-priority target for the temperature at the start of simulation. To enable this variable, in the Thermal section, select the Model thermal port parameter. (since R2024a)

Examples

AM Radio Receiver

A simplified AM radio receiver. A single tone signal at 2kHz is transmitted with a carrier frequency of 600kHz. The variable capacitor, Cres, in the resonant circuit is used in order to sweep through a certain frequency span. When the resonance passes through 600kHz, the signal is picked up and amplified by a two-stage Class A RF power amplifier. The signal is finally extracted by a diode detector, where it would normally be passed on to an audio amplifier (not included here). The Scope displays the final output, the value of the resonant capacitance, and the received and amplified signals.

• Open Model

Ports

Conserving

+ — Positive terminal

electrical

Electrical conserving port associated with the capacitor positive voltage.

- — Negative terminal

electrical

Electrical conserving port associated with the capacitor negative voltage.

H — Capacitor thermal mass

thermal

Since R2024a

Thermal conserving port associated with the capacitor thermal mass.

Parameters

Main

Capacitance model — Capacitor type

Constant (default) | Lookup table | Dielectric relaxation (Debye)

Select the type of capacitor:

• Constant — Model a linear capacitor, with nominal capacitance defined by the Capacitance parameter value.

• Lookup table — Model a nonlinear capacitor, where the nominal capacitance value changes based on the value of applied terminal voltage.

• Dielectric relaxation (Debye) —Model a frequency-dependent capacitor with resistive and dielectric losses.

Capacitance — Nominal capacitance

1e-6 F (default)

The nominal capacitance value for linear capacitor, or the low-frequency capacitance in the Debye parameterization.

Dependencies

This parameter is visible only when you select Constant or Dielectric relaxation (Debye) for the Capacitance model parameter.

Capacitance values — Vector of capacitance values

[1e-05, 1e-06] F (default) | positive vector

The vector of capacitance values, for table lookup based on the corresponding voltage value. The vector length must be the same as the voltage vector length.

Dependencies

This parameter is visible only when you select Lookup table for the Capacitance model parameter.

Corresponding voltage values — Input vector of voltage values

[0, 10] V (default)

The input vector of voltage values for table-based capacitance calculation. The vector length must be greater than or equal to 2, and the values must be strictly monotonic, either increasing or decreasing.

Dependencies

This parameter is visible only when you select Lookup table for the Capacitance model parameter.

Symmetric C-V table — Table data

Yes - use voltage magnitude when computing C (default) | No - use C-V data as-is

Specify how to use the table data:

• Yes - use voltage magnitude when computing C — Use this option to ensure symmetry of the capacitance with regards to the applied terminal voltage.

• No - use C-V data as-is — Use this option to model polar capacitors. For example, with default parameter values for table-based capacitance, applied voltage of –10 V would produce nominal capacitance of 1e-6 F. However, if you select No - use C-V data as-is for the Symmetric C-V table parameter, the resulting capacitance value is 1e-5 F, because the block uses the nearest input value for extrapolation.

Dependencies

This parameter is visible only when you select Lookup table for the Capacitance model parameter.

Frequencies for specifying dissipation factors [f1 f2] — Frequencies for dissipation factors calculation

[1, 10] kHz (default)

Frequencies at which the dissipation factors [DF1 DF2] are calculated, in kHz.

Dependencies

This parameter is visible only when you select Dielectric relaxation (Debye) for the Capacitance model parameter.

Dissipation factors (%) at f1 and f2 [DF1 DF2] — Dissipation factors at f1 and f2

[.8, 1.2] (default)

Ratio between the equivalent series resistance and the capacitive reactance, or the tangent of the loss angle. Dissipation factors are a common metric for capacitors.

Dependencies

This parameter is visible only when you select Dielectric relaxation (Debye) for the Capacitance model parameter.

Capacitance tolerance (%) — Capacitor tolerance

5 (default)

The capacitor tolerance as defined on the manufacturer datasheet. For table-based capacitors, this tolerance is applied to the entire table at once.

Tolerance application — Tolerance application

None — use nominal value (default) | Random tolerance | Apply maximum tolerance value | Apply minimum tolerance value

Select how to apply tolerance during simulation:

• None — use nominal value — The block does not apply tolerance, uses the nominal capacitance value.

• Random tolerance — The block applies random offset to the capacitance value, within the tolerance value limit. You can choose Uniform or Gaussian distribution for calculating the random number by using the Tolerance distribution parameter.

• Apply maximum tolerance value — The capacitance is increased by the specified tolerance percent value.

• Apply minimum tolerance value — The capacitance is decreased by the specified tolerance percent value.

Tolerance distribution — Tolerance distribution type

Uniform (default) | Gaussian

Select the distribution type:

• Uniform — Uniform distribution.

• Gaussian — Gaussian distribution.

Dependencies

This parameter is visible only when you select Random tolerance for the Tolerance application parameter.

Number of standard deviations for quoted tolerance — Number of standard deviations for quoted tolerance

4 (default)

Number of standard deviations for calculating the Gaussian random number.

Dependencies

This parameter is visible only when you select Gaussian for the Tolerance distribution parameter.

Series resistance — Series resistance

1e-6 Ohm (default)

Simulation of some circuits may require the presence of the small series resistance. Equivalent series resistance (ESR) is sometimes specified on manufacturer datasheets. If not, you can define this resistance for a linear capacitor via the dissipation factor (DF), which is also shown on many datasheets. The relationship is DF = 2π· f· C· ESR, where f is signal frequency. For a Debye capacitor, the Dissipation factors (%) at f1 and f2 [DF1 DF2] are corrected for this additional series resistance prior to computing 𝛼 and 𝜏.

Parallel conductance — Parallel leakage path

0 1/Ohm (default)

Parallel leakage path associated with the capacitor. For capacitors connected in series, the presence of a small parallel conductance can help with convergence.

Averaging period for power logging — Averaging period for power logging

0 s (default)

Averaging period for power logging, in s.

Note

If this parameter is set to 0, the output is the instantaneous power.

Dependencies

This parameter is visible only when you select Dielectric relaxation (Debye) for the Capacitance model parameter.

Operating Limits

Enable operating limits — Enable operating limits

No (default) | Yes

Select Yes to enable reporting when the operational limits are exceeded. The associated parameters become visible on the Operating Limits tab to let you select the reporting method and specify the operating limits in terms of power and working voltage.

Reporting if operating limits exceeded — Report behavior

Warn (default) | Error

Select what happens when an operating limit is exceeded:

• Warn — The block issues a warning.

• Error — Simulation stops with an error.

Working voltage range — Working voltage range

[-25, 25] V (default)

Range of voltage values allowed for normal block operation, specified as a vector of size 2.

Power rating — Maximum instantaneous power dissipation

1 W (default)

Maximum instantaneous power dissipation in the resistance and conductance elements associated with the capacitor.

Temperature — Maximum permissible temperature

85 degC (default)

Since R2024a

Maximum permissible temperature. If the temperature of the capacitor is greater than this value, the block throws a warning or an error.

Dependencies

To enable this parameter, in the Thermal section, select the Model thermal port parameter.

Thermal

Model thermal port — Option to model thermal port

off (default) | on

Since R2024a

Option to model a thermal port in the Capacitor block.

Thermal mass — Thermal mass associated with thermal port

1 J/K (default)

Since R2024a

Thermal mass associated with the thermal port H. It represents the energy required to raise the temperature of the thermal port by one degree.

Dependencies

To enable this parameter, select the Model thermal port parameter.

Faults

Capacitor fault — Option to model fault in capacitor

Add fault (default)

Option to model a fault in the Capacitor block.

To add a fault, click the Add fault hyperlink.

Faulted capacitance as % of unfaulted — Faulted capacitance as % of unfaulted

100 (default) | positive scalar

Relative change in the capacitance when the block is in the faulted state, as compared to the unfaulted state. For table-based capacitances, the relative change is applied to all elements of the vector. The default value is 100, which means that the faulted capacitance is equal to the unfaulted capacitance.

Dependencies

To enable this parameter, add a fault to the Capacitor block by clicking the Add fault hyperlink in the Capacitor fault parameter.

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

Faulted series resistance — Faulted series resistance

1e-3 Ohm (default) | finite positive scalar

Equivalent series resistance of the capacitor when the block is in the faulted state.

Dependencies

To enable this parameter, add a fault to the Capacitor block by clicking the Add fault hyperlink in the Capacitor fault parameter.

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

Faulted parallel conductance — Faulted parallel conductance

0 1/Ohm (default) | finite nonnegative scalar

Parallel leakage conductance of the capacitor when the block is in the faulted state.

Dependencies

To enable this parameter, add a fault to the Capacitor block by clicking the Add fault hyperlink in the Capacitor fault parameter.

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

Fault transition time constant — Fault transition time constant

1e-3 s (default) | positive scalar

Time constant associated with the transition to the faulted state, as described in Faults.

Dependencies

To enable this parameter, add a fault to the Capacitor block by clicking the Add fault hyperlink in the Capacitor fault parameter.

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

Trigger fault at time — Time before entering faulted state

0 s (default) | nonnegative scalar

Simulation time at which the block enters the faulted state.

Dependencies

To enable this parameter, in the Fault Inspector window, set Trigger Type to Timed.

This parameter appears in the Trigger section of the Fault Inspector window. For more information, see Set Fault Triggers.

Maximum permissible temperature — Maximum permissible temperature

85 degC (default) | nonnegative scalar

Since R2024a

Maximum permissible temperature. If the temperature value is outside this value for longer than the Time to fail when exceeding maximum temperature parameter value, then the block starts entering the fault state.

Dependencies

To enable this parameter, in the Fault Inspector window, set Trigger Type to Behavioral.

This parameter appears in the Trigger section of the Fault Inspector window. For more information, see Set Fault Triggers.

Time to fail when exceeding maximum temperature — Time to fail when exceeding maximum temperature

1 s (default) | nonnegative scalar

Since R2024a

Maximum length of time that the temperature can be outside the permissible temperature range without triggering the fault.

Dependencies

To enable this parameter, in the Fault Inspector window, set Trigger Type to Behavioral.

This parameter appears in the Trigger section of the Fault Inspector window. For more information, see Set Fault Triggers.

Permissible voltage range — Minimum and maximum permissible voltage

[-100, 100] V (default) | vector of length 2

Minimum and maximum permissible voltage. If the voltage value is outside this range for longer than the Time to fail when exceeding voltage range parameter value, then the block starts entering the fault state.

Dependencies

To enable this parameter, in the Fault Inspector window, set Trigger Type to Behavioral.

This parameter appears in the Trigger section of the Fault Inspector window. For more information, see Set Fault Triggers.

Time to fail when exceeding voltage range — Time to fail when exceeding voltage range

1 s (default) | nonnegative scalar

Maximum length of time that the voltage can be outside the permissible voltage range without triggering the fault.

Dependencies

To enable this parameter, in the Fault Inspector window, set Trigger Type to Behavioral.

This parameter appears in the Trigger section of the Fault Inspector window. For more information, see Set Fault Triggers.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2016a

R2024a: Model thermal effects by exposing thermal port

The Capacitor block now supports a thermal port. To model a thermal port, in the Thermal section, select the Model thermal port parameter. To specify the value of the thermal mass and the maximum permissible temperature of the capacitor, specify the values of the new Temperature and Thermal mass parameters in the Operating Limits and Thermal sections, respectively.

R2023a: Implement temporal or behavioral triggers for faults using a single parameter

The Enable temporal trigger and Enable behavioral trigger parameters have been removed and are replaced with the Fault trigger parameter. You can set the Fault trigger parameter to Temporal or Behavioral. You can enable only one trigger.

If you created a model in an earlier release and enabled both temporal and behavioral triggers, now only the temporal trigger is enabled. Therefore, the simulation results are different if the behavioral trigger causes the fault in your model.

If you want to trigger a fault at a specific time, set the Fault trigger parameter to Temporal. If you want to determine whether a system fails and, if so, when it fails, set the Fault Trigger parameter to Behavioral.